- Email: [email protected]
Stretch-bend coupling effect on the intramolecular vibrational relaxation in a two-mode model of CH overtone excited fluoroform
Chemical Physics 141 ( 1990) 197-209 North-Holland
STRETCH-BEND COUPLING EFFECT ON THE INTRAMOLECULAR VIBRATIONAL RELAXATION IN A TWO-MODE MODEL OF CH OVERTONE EXCITED FLUOROFORM
Andms GARCfA-AYLLGN and Jesus SANTAMARfA Departamento de Q&mica Fisica, Fact&ad de CC. Q&micas, Universidad Complutense, 28040 Madrid, Spain Received 28 Match 1989; in final form 10 October 1989
A classical dynamic study of the effect of stretch-bend coupling and CH anharmonicity on the occurrence of Fermi resonance and intramolecular vibrational relaxation (IVR) has been carried out for a two-mode Hamiltonian proposed by Halonen et al. for CH overtone excited fluoroform. The analysis is made in terms of variations of the phase space structure and overtone relaxation rate constants, with respect to CH stretch anharmonicity and various approximations to the kinetic and potential couplings. CH bond anharmonicity is confirmed as the most important factor in the dynamics of overtone relaxation and it controls the occurrence and intensity of the 2 : 1 Fermi resonance between the CH stretch and the adjacent bends. The kinetic coupling is much more important than the potential, and, contrary to the effect of this, it tends to decrease the stability of the CH periodic orbit. The overtone relaxation rate constants, estimated by linear stability analysis of the CH periodic orbit, are in good agreement with the experimental results.
1.
Introduction
The advent of new experimental techniques, such as overtone spectroscopy has provided information about molecular behavior at large amplitude vibrational motions. Recently there has been a large amount of experimental research on the overtone spectroscopy of CH and OH local modes [ l-6 1. The dynamics of coupled vibrations is of capital importance in the understanding of molecular spectra and intramolecular energy transfer. At the same time the presence of low-order frequency commensurability between anharmonic zeroth-order modes (i.e. the generalized Fermi resonances) is the cause of the intramolecular vibrational relaxation (IVR) and splittings in the vibration-rotation spectra [ 71. Assuming that the observed linewidths are due to homogeneous broadening, the time scale of the initial vibrational relaxation can be inferred [ 8,9]. In each case the overtone relaxation is found to occur within a fraction of a picosecond [ lo]. Much recent theoretical work [ 11-211 has been focused on the extraction of information from overtone spectroscopy on the specific pathways that govem IVR. Of particular importance is the coupling between CH (or OH ) stretching vibrations and adjacent 0301-0104/90/$03.50 ( North-Holland )
0 Elsevier Science Publishers B.V.
bending modes. The initial energy transfer from the excited stretch to the adjacent modes has been shown to proceed via a 2 : 1 Fermi resonance whose magnitude varies with the specific nature of each molecule. Several resolved absorption bands appear in each overtone region in trihalomethanes [ 2-51. These bands result from a very strong coupling between the CH stretch and the bends and are characteristic of Fermi resonances. The measured splitting within each polyad allows the stretch-bend anharmonic coupling to be determined quantitatively, and demonstrates that CH stretching energy flows first to the bending modes and, subsequently, to the heavy atom vibrations on a longer time scale. It has been shown [ 2-5 ] that these Fermi resonances in CHFs, CHD, and CH( CF,)3 can be analysed via effective tridiagonal Hamiltonian matrices. The diagonal terms contain both harmonic and anharmonic contributions, while the offdiagonal Fermi resonance coupling matrix elements depend on an effective coupling constant (KW). The use of curvilinear internal coordinates and their appropriate symmetrized combinations [ 7,16,22,23 ] represents clear advantages over the Cartesian normal mode coordinates for the large amplitude vibrational motions of the overtone excitations. First of
198
A. Garcia-Aylldn, J. Santamaria / A two-mode model ofjuoroform
all, it accounts for the anharmonicity of X-H bonds; secondly, curvilinear coordinates give a more accurate representation of the potential energy surface when compared with rectilinear coordinates, and, tinally, the potential surface parameters are isotope independent within the Born-Oppenheimer approximation. A problem with these coordinates is the fact that the G matrix elements of the kinetic energy are much more complicated than in the case of rectilinear coordinates. The study of Fermi resonances between CH stretching and HCF bending vibrations of fluoroform has been carried out by Halonen et al. [ 171, using a Hamiltonian described in terms of symmetrized curvilinear internal coordinates and their conjugate momenta. The kinetic energy operator is obtained by expanding the G matrix in terms of the Morse function y = 1 - exp ( - PAr ) , and retaining only the relevant terms for the Fermi resonance problem. The potential energy function is also expanded about the equilibrium configuration in terms of y. Symmetrized internal combinations of curvilinear coordinates were used to reduce the problem to a two-mode (HC stretch and a doubly degenerate bend) model Hamiltonian. These coordinates were primarily used by Voth et al. [ 151 in their theoretical analysis of the highly excited CH vibration in partially deuterated methanes. In this study, the G matrix elements of the Hamiltonian were evaluated with all internal coordinates at their equilibrium values, except the CH coordinate r. In addition to these G matrix couplings, higher-order (i.e. higher than quadratic) curvilinear potential energy terms were needed in order to explain more quantitatively the observed spectra of deuterated methanes. Halonen et al. [ 171 use the same type of approach but they expand the kinetic energy operator in terms of y. The model Hamiltonian they use gives band positions and relative band intensities within polyad spectra with enough accuracy. At the same time, the description of Fermi resonance is made in terms of force constants which have a precise definition. A detailed dynamical treatment, but restricted to the effective resonance Hamiltonian spectroscopically fitted, has been carried out by Kellman [ 19 1. The underlying idea of this approach, so-called algebraic resonance dynamics, is that every element of the spectroscopic fit corresponds to an element of the non-linear dynamics. The polyad phase space pro-
files obtained by Kellman for CHX3 molecules render a phase space structure very different from the simple pendulum resonance Hamiltonian model 1241. In this paper, we present a classical dynamical treatment of the effect of both kinetic and potential couplings between the CH stretch and the doubly degenerate bend for several variations of the two-mode model Hamiltonian proposed by Halonen et al. [ 17 ] for CHF3 molecule. Following the treatment used in our study of a two-mode model of benzene [ 13 1, we are trying to understand the sensitivity of short time IVR to potential and kinetic couplings in terms of the classical phase space structure of the model Hamiltonian for fluoroform. The change from stability to instability (and vice versa) of the anharmonic CH periodic orbit and the variation in intensity of the Fermi resonance between the stretch and the bends are a function of potential and kinetic coupling as well as stretch and bend anharmonicities. It is our goal to study the changes in phase space structure (and Fermi resonance intensity) and the variation of overtone relaxation rate constants with respect to various approximations of both kinetic and potential coupling terms of the two-mode Hamiltonian. The various versions of the Hamiltonian are described in section 2, where we discuss the approximations we use. In section 3, the Poincart surface of section is used to examine changes in phase space structure induced by the stretch-bend potential and kinetic energy couplings. Section 4 is devoted to estimate the short time overtone relaxation by the method of linear stability analysis. In section 5 we present the conclusions and discuss possibilities for future work.
2. Model Hamiltonian The reduced classical Hamiltonian used to study the Fermi resonances between the CH stretch and the adjacent HCF bending vibrations in fluoroform CHCFJ is described in terms of internal coordinates and conjugate momenta as follows (see fig. 1):
A. Garcia-Ayllbn, J. Santamaria / A two-mode model offluoroform
199
where t,uis the dihedral angle defined by (Yand #. Taking advantage of the symmetry of the problem, the Hamiltonian (2.la) is transformed to symmetrized internal coordinates [ 17 ] ‘%= 3-l”
(pi
+v)2
6 = 6-“‘2 (2% 62=
Fig. 1.Molecular model of CHF3.
2-l’*
+cQjh
-92
-$@3),
(2.3a) (2.3b) (2.3~)
(v)z-g3),
and their conjugate momentap,, p6,, plh. If the totally symmetric bending coordinate 0, is fixed at its equilibrium value, then its conjugate momentum ps disappears, and the Hamiltonian takes the form
+ Ge,& pet P& + i=q 2 Grei PC~ei +D[
1-exp( -/3Ar)12+ V(r, 8,, tl,),
(2.4)
where V(r, 8,, 6,) can be taken as a Taylor expansion similar to (2.1 b) but now in terms of or, S, and r. The new G matrix coefficients are
(2.lb) Here (r, p,) are the coordinate and momentum of CH stretch, and (pi, pe i= 1, 2, 3) are the three HCX bending vibrations. The CH stretch is described by a Morse oscillator and the bends can be considered as harmonic oscillators with r-dependent force constants as we will show later on. The CF bond stretch coordinates and the FCF bond angle a! are frozen at their equilibrium values (R=1,332~,anda=108” respectively,seetig. 1). The kinetic energy G-matrix elements are [ 25 ] (2.2a)
G,=lImH+lImc,
- 2 cos &rR)/mc,
G,j = -sin ~i~Rm~,
[ cos
W/r2- 2 cos
(-G22+G33+G,2-G,3),
G&,= 6-1’2 (2G,, -GR-Gr3), G&z
2-lf2 (Gr2-G,3).
(2.5)
The approximation made following Voth et al. [ 15 ] is to treat the dynamical G matrix elements in eqs. (2.2) as functions of the CH stretch coordinate exclusively, with the rest of the variables (R, pi, v/) at their equilib~um values (see table 1) . Consequently the G matrix elements in terms of pior B are G22
=G33
=G,(r),
(2.2b)
GEz=G t3=G23=Gwstr),
(2.2c)
G,,,,=G,,=G,(r)=G,(r)-Go,, GBI&-0 9
Gti=cos ylr2mH +
G@,&= 3-l”
GI I =
Gii = 1/r2mH+ 1/R2mc +(l/r2+1/R2
G.a=t~22+fG33-G23,
c;Oicos
G,.o,=Gti=G,+=O.
y/rR
+ (sin’pli sin2yl+cos (Ycos y)/R2]/mc,
(2.2d)
(2.6)
Since this is the most obvious simplification of the
200
A. Garcia-Aylh. J. Santamanh /A two-mode model ofJluoroform
Table 1 Surface parameters
H= ;G,p;+; Ab initio
f K) &A) a=~+-
(deg)
P=PW
(deg)
Fit
39386 1.861 1.098 I .332 108.0 110.13 1.008 18.998 12.000 44540 - 18997 - 88895
D (cm-‘)
mH (au) mF (au) mc (au)
(cm-’ radm2) F, (cm-’ A-’ radm2) F,, (cm-’ A-* rade2) F
jG,
42310 1.801
+D[
40854 10550 - 284000
H=tG,,p:+fGBepi+D[l-exp(-j?Ar)]’
J
P;
V(r, 8,, 0,).
The effect of the cross term in p,pe is very small due to the small numerical values of G2 given by Halonen et al. [ 171. With respect to the angle dependence of the potential function V( r, 0,, 0,) it is convenient to take the bending vibrations as harmonic oscillators with r-dependent force constantfO( r) in the form:
= t LLi(rM+Mrb%l
= i.L4r)e2,
fe(r)=Fw+
Halonen et al. [ 17 ] have carried out an expansion of the “quasiexact” Geeand Gti matrix elements in terms of the Morse function y= 1-exp( - /IAr) and of 0, respectively, resulting in
Gti,==G$i+G2Oi
f (i=l,
>
$G3+fGl
~J’,wY
2)y
(2.8)
Ar=O, p=O), GS = (aZG&r2)er
(2.11)
Alternatively this is easily verified by expanding the potential function in (pand 8 variables, Fee =fap--fwf>
Frm=fw-fw,, Fwee=fr,,-f-p.
Y2,
where
G, = (6IGeel%)e,
(2.10)
with e2= f3: + 0:. The r dependence of fe is easily found by expanding in terms of y ( Morse function )
y2.
jG,y+
(2.9)
12.7)
e2),
P; =PW +&.
G&=G,(
>
Y’
1 -exp( -/3Ar)]‘+
with
G,=G&+
1
1
problem, we will consider the expressions for GBBand G,+given in (2.6) as quasiexact. The classical Hamiltonian now takes the form
+ Ur, 4,
G&+ ;GY
(2.12)
Here the f symbols stand for the coefficients up to fourth order in the expansion of the potential in r and $ displacements. In this way we have an analytical expression for the stretch-bend potential coupling in terms of a switching function S(y) of the form Ur, 0) = t%Y)&@,
G~,=G~=G~(Ar=O,yl=O)=O, G2 = (aGJ@),. In this approximation the classical Hamiltonian takes the form
(2.13) The switching function in (2.13) is a parabola in y which depends on the values of the potential expan-
201
A. Garcia-Ayll&n,J. Santamaria / A twcFmodemodel offluoroform
sion coefficients. Two series of coefficient values are given by Halonen et al. (see table 1 ), the first coming from ab initio calculations [ 26 1, and the second obtained by fitting the values of FM and F,* (and consequently the parameters X, and Ksbbof the diagonal and tridiagonal elements [ 21) to the experimental spectrum up to the chromophore quantum number of the polyad N= n,+ in@= 6 (for A, states) and 1 1/ 2 (for E states). The analytical expressions for S(y) for both series of parameters are S(y) = l-O.229 ly- 0.4027y2,
(2.14a)
S(y) = 1+0.1434y-o.9999y2,
(2.14b)
respectively, and the representations are given in fig. 2 within the range from y= - 1 (Ar= -0.37 A) to y= 1 (At= + 00). Both functions differ considerably for Ar < 0, and at the same time, neither give the correct asymptotic behavior at y= 1, S( 1) =O. In order to correct this anomaly we have used an analytical expression of the form [ 27 ] S(y) = 1-tanh(P&) (2.15)
=2(1-y)*‘/[l+(l-y)2c],
which gives the asymptotic behavior and passes for (co.3 16 through the two crossing points of the switching functions previously indicated (see fig. 2). I
-
Ab m,t,o
-,-
Asymptotac
In fig. 2 the classical turning points of the CH Morse function for n,= 4 and 8 are also shown. In summary, we have different expressions for the two-mode (CH stretch and a doubly degenerate bend) model Hamiltonian, which include two forms ( (2.7) and (2.9) ) for the kinetic energy and four for the potential coupling: two coming from (2.13) with “ab initio” and “fitted” potential parameters respectively ( 2.14 ) , the asymptotic expression ( 2.15 ) and finally the case of no potential stretch-bend coupling, i.e. S(y) = 1. All possible combinations allow us to study in depth the effects of changing contributions of the kinetic and potential coupling on the stability of HC periodic orbit and on the intensity of 2 : 1 Fermi resonance.
3. Stretch-bend potential, kinetic couplings and phase space structure We have used surfaces of section [ 281 to study the phase space structure of the two-mode model Hamiltonian for the various versions previously described. Each surface of section (SOS) is obtained by plotting (f3,pe) every time r= r. and pr> 0 for several trajectories at a particular constant energy and different initial actions. The gross features of the phase space corresponding to a system of two degrees of freedom with the possibility of 2 : 1 Fermi resonance are already well known [29,30,13]. In our system, however, the anharmonic stretch-bend resonance is known to be small for the stretching fundamental and the first two overtones, but the large Morse anharmonicity of the CH stretch can enhance the resonance. In fact, the Morse frequency o, is defined as o, =a( l -&!1,/2D),
(3.la)
where Q is the harmonic frequency, 9= (2DB* x Grr)“‘, and Z, is the Morse action, while the bend harmonic frequency oh with an r-dependent force constant S( y ) Fm has the expression we= ]GBBS(Y)J’~~I”~,
r.-0.37
A
Y Fig. 2. Switching fimctions S(y).
(3.lb)
with the result that w, can be larger, smaller or approximately equal to two times o& Representative SOSs for quasiexact Gee and asymptotic potential coupling ( 2.15 ) are shown in fig. 3 at energies corresponding to initial states (n, ne=O), n,=2, 3, 5 and
202
A. Garcia-Ayllbn. J. Santamaria /A two-mode model ofJluoroform
3.00
liil,i,,,,,,,,,,,,,,,,,l,,,,,,,,,,,,l,,,,,,,
d i
Fig. 3. SOS for Hamiltonian (2.7) ( and asymptotic S(y) at energies corresponding to overtone excited states (n, ne). Representation ofp, (in amu A/IO-” s) versus 19(in rad). (a) State (2,0), (b) state (3,0), (c) state (5,0), (d) state (10, 0).
10. At n,= 2, the structure of SOS is very simple with a stable (elliptic [ 281) central fixed point, because we are far from resonance. In fig. 3a the SOS is not shown in full: a more detailed representation should give us two small islands at the top left and bottom right [ 291, such as are observed for nr= 5 (fig. 3~). There is a single family of regular trajectories (apart from microscopic regions of localized stochasticity which are always present in non-integrable systems [ 281) continuously spanning the range between two
limiting periodic orbits: the pure CH stretch motion, corresponding to the central fixed point in the SOS, and the periodic orbit piercing the SOS at the center of each of the two islands [ 131. Fig. 3b shows the SOS of the lower energy level in which the central fixed point is unstable (hyperbolic with reflection [28] ) with two elliptic points clearly visible. Obviously we are getting close to the resonance. Again the two islands mentioned in the previous cases are not shown. The situation is more clear in fig. 3c where
A. Garcia-Ayl~~m~ f. Santamark /A two-mode modei of~uor~forrn
there are two families of trajectories divided by the separatrix of the resonance: one resonant family (small region inside the separatrix) with large interchange of energy between the stretch and the bend, and other non-reson~t family (large region outside the separatrix). A representative trajectory of the resonant family is drawn in fig. 4a. The two islands at the top left and
1
.oo i
/
a
$
500
-f.OO C
j
E
0.00 ?
3 -1.50
m--lo
0.00
1.25 CH
Fig. 4. Can~guration space plots. Rep~~n~tion off? (in rad) versus r (in A). (a} Representative trajectory of the resonance family at the (4,O) state for Hamiltonian (2.7) and asymptotic S(y). (b) Same for state (8,O).
203
bottom right of the SOS of fig. 3c correspond to a single trajectory of the non-resonant family, and are a consequence of the particular cut used to define the SOS [ 13 1. At n,= 7 and 8 we are exactly in resonance (with an hyper~lic central fixed point and four elliptic ones) and, as a result of the bifurcation of the CH stretch periodic orbit, we have two families of resonant trajectories divided by a unique separatrix (see fig. 5d). The first family is delimited by the two elliptic points at the top right and bottom left and by the separatrix, and representative trajectories have the form of ( (open right) in the configuration space plot (r, 8), similar to fig. 4a. They correspond to the situation where stretch elongation and angle increment are in phase. The second, also resonant family, delimited by the two other elliptic points and the separatrix, define trajectories where stretch elongation and angle variation are in phase opposition. Representative trajectories have the form of > (open left) (see fig. 4b). As it is clear from fig. 5d, the ( family gives more interchange of energy between stretch and bend due to the different variation ofp, in the two limiting stable trajectories piercing the SOS at their couple of elliptic points. When the energy of CH increases (higher overtones) the system goes out of resonance due to the CH strong anharmonicity. For example, at nr= 10 (fig. 3d) we have a situation similar to the case n,=4 with two families of trajectories (resonant and nonresonant), but now the resonant family includes the trajectories of the form of) (i.e. with r and 8 motions in phase opposition). On the other hand, the correlation between the cases n,=5 and t2,= 10 cannot be complete, because in the last case we are approaching stochasticity. As we mentioned in section 2, four types of potential coupling have been investigated. The effect of the potential coupling is not very severe within the range of bend attenuation studied. There are, nevertheless, significant differences which deserve some attention. As is well known [ 12,13 ] the potential coupling reduces the range (in energy) of instability in the CH stretch periodic orbit. Within the range of energies explored (0 < n,< lo), the strongest potential couplings correspond to the cases of asymptotic and ab initio switching functions, because they give the smallest resonance width with the closest distance in phase space between the two elliptic resonant points (see fig. 5a for n,=4). The asymptotic switching
-
2
31
1
I
/
,
I
,
,
,
a
0.00
-3.00 -;
2
0.00 -
-3.00
pi -2.00
0.00
2 00
3.00
Fig.S.SOSforHamiltonian (2.7).Representationofp~(inamu~2/10-’4s)versusB(inmd). (a) State (4,0)andabinitioS(y), (b) state (4,O) and fitted s(y), (c) state (4,O) and no coupling, (d) state (8,O) and ab initio S(y), (e) state (8,O) and fitted S(y), (f) state ( 8,0 f and no coupling.
A. Garcia-Ayllbn, J. Santamaria / A two-mode model offluoroform
function yields a slightly lower instability than the ab initio function. The empirically fitted attenuation function (fig. 5b) offers an anomalous behavior, in the sense that, although the resonance width is lower than in the case of no coupling (and larger than in the two previous cases), see fig. 5c, nevertheless, the pure resonant trajectory gives more interchange of energy (i.e. larger distance between elliptic points than in the no coupling situation). The reason is due to the fact that S(y) can be greater than unity for values of y slightly positive, while for y
205
0
2 0.00-
-3.00 !I -2.00
. ‘., I’,:i : .‘.J,y-“‘i .\ .I~ ;.,>..,‘., , .,, ‘f ., ::.: +_.,__ :,+ *,.,, .’1;:;.:...‘,:. : .‘i . “.“’ ’ : ,i : . . .‘% . . I ._... -I :: ‘..T..::: 4 ” ...s .,::...i -.<..’ . : _... ,:,.-... ,i, ,. :.:A:,.., .., . .._ I.. . ;f,,‘,” - “.$$ .i’ :, ‘:,‘. “..:,... >:.,.. : . +.;..,,,.i:;. /X!‘, -...I,_
000
2 00
19
Fig. 6. SOS for Hamiltonian (2.9) with ab initio potential coupling at different overtone excited state (n, ne) . Representation ofp, (in amu A2/10-” s) versus 0 (in rad). (a) State (4, 0), (b) state (8,O).
4. Short time overtone relaxation and linear stability analysis The short time decay rate constants from the CH overtones in the two-mode model can be estimated very accurately by linearization of the dynamics [ 28 ]
A. Garcia-Ayiih, J. Santamarfa /A two-mode model offluoroform
206
around the HC periodic orbit [ 13,321. Linear transformations are area preserving and there is a Jacobian (2 x 2 ) matrix propagator whose determinant is unity. The eigenvalues of the Jacobian matrix propagator of the return map are I, =exp( ia), R,=exp( -irr) for elliptic (stable) fixed point, and A,=exp(cr), Az=exp(-o) for hyperbolic (unstable) fixed point (D real), where the exponents are the (real or complex) phases acquired by the eigenvectors after 4 (the period of the orbit). In the unstable case it is possible to estimate the decay constant k for the stretch-bend relaxation, since then rr=l~7~,. The method requires the numerical calculation of the Jacobian matrix by determining the action of the return map on initial conditions displaced by small amounts (Ape, AfI=O.OOl ) along the 0 and pe directions respectively. This is easily achieved by running for one period r, a couple of trajectories, with initial conditions displaced the first trajectory in pB _ and the second in 0. The relaxation rate k is estimated by diagonalizing the resulting Jacobian matrix. The values of k (ps-’ ) for all the situations discussed in section 3 are given in table 2. The comparison of the values of k confirms all the deductions we made in terms of phase space structure. Only the homogeneous widths (r,,,) can be associated with the decay of the population for initially prepared CH stretching and bending states (n,, n,; nr= 1, .-‘> 10, ne=O). For CHF, the exponential decay time 7= 1/2~cr,,~, was estimated, using the
time evolution matrix method [ 21, to be about 0.1 ps for the state n,= 3. The value we get for the same initial state is about 0.07 ps (r=75 cm-’ ), and for n,=8 t is of the order of 0.04 (r= 130 cm-‘). For CH f CF3) 3 Baggott et al. [ 3 ] gave rough estimates for lY which are between f= 7 cm-r for the N= 2 polyad and r=70 cm-’ for the N=6 polyad, these widths are related to decay times which range between 0.8 and 0.08 ps. Our results are reasonable and fall within the experimental uncertainties. It is interesting to note the slight effect of the potentiai coupling on the stability and resonance of the periodic CH orbits. In fact, the suppression of the coupling displaces the resonance toward lower overtones (from n,= 7, 8 toward n,= 5, 6 ) and increases the decay constant by less than a factor of two. Much more pronounced changes appear by suppressing the kinetic coupling, i.e. taking r equal to (r) of each overtone level in the Gee expression (G( (r) ) zconstant). In that situation the instability and resonance are only present within the range 2
Table 2 Rate constants (ps-’ ) for overtone decay in model Hamiltonian for fluoroform from linear stability analysis %
I 2 3 4 5 6 7 8 9 10
Fitted S(y)
Asymptotic S(y)
Ab initio S(y)
No potential coupling
G, approx.
GMquasiexact
GBB constant
GIXI approx.
GBequasiexact
GBB approx.
Gwquasiexact
Gm approx.
GBBquasiexact
n a) 4 16 21 23 24 23 21 18 12
n n 14 19 22 23 24 23 21 18
n 7 12 11 n n n n n n
1: 25 30 32 33 34 33 32 30
n 16 24 28 31 33 33 33 33 32
n 5 17 22 25 27 27 26 24 20
n n 15 21 24 26 27 27 26 24
16 27 34 37 38 38 35 30 21 n
14 26 33 36 38 38 37 34 29 21
a1 n denotes non-exponential decay.
A. ~ar~ia-Ay~l~~,.I. Santamaria / A two-mode model of~~oroform
effects work in the opposite direction: the potential coupling produces stability of the CH periodic orbit, while the kinetic renders the system more dynamically unstable. A quantitative study of those effects is under way [ 331, where we treat the stretch-bend resonance Hamiltonian by Fourier expansion, via the analytical evaluation of the appropriate Fourier coefficients [ 13 1. We have also investigated the question of heavy atom blocking on stretch-bend vibrational energy transfer for this model of fluoroform. Working with infinite mass for carbon atom, for fluorine atom or for both atoms, the modifications on the stability decay of CH periodic orbit in comparison with the case of real atom masses are very small as one can see in table 3. This is not surprising since the specific nature of a bend motion prevents an easy blocking of energy transfer from the adjacent stretch.
5. Discussion and remarks In this work we have examined the sensitivity of classical dynamics of a two-mode model Hamiltonian for fluoroform. By examining the classical phase structure of the two-mode system, we have shown that there is a strong correlation between exponential short time decay constant by using linear analysis and stability of the CH stretch periodic orbit as observed in the phase space structure. In a previous work [ 131 we showed that the values of the relaxation rate con-
stant obtained by linear stability analysis were in good agreement with the quasiclassical trajectory results as measured from the probability P*(t) for remaining in the initial state (n, 0). This last finding, which we assume to be general, dispenses with running batches of quasiclassical trajectories in this study. The effect of CH bond anha~onicity on the stability and resonance of the CH periodic orbit has been shown to be, as in similar systems [ 1l-l 3 1, extremely important. On the other hand, the effect on the dynamics as coming from changes in the potential coupling is not very strong within the explored range of stretch-bend attenuation. One would need a substantial change (much more strong attenuation [ 12 ] ) to produce significant changes on the dynamics, such as changes from instability to stability or from resonance to out of resonance of the periodic orbits. Nevertheless the effects are noticeable and one can correlate the increase of potential coupling attenuation with the increase of stability of the CH periodic orbit. The suppression of the kinetic coupling produces a dramatic effect on the dynamics and phase space structure of the two-mode system. Again one can correlate, contrary to the potential coupling effect, the increase in strength of the kinetic coupling with the decrease of stability of the orbits (and with the displacement of the resonance to lower overtones ), We have also confirmed that the heavy atom blocking effect does not exist for stretch-bend vibrational energy transfer. This situation is nevertheless
Table 3 Heavy atom blocking effect on the rate constants fps-’ ) for overtone decay in model Hamiltonian of ~uo~fo~. exact and asymptotic S’(y). Masses of carbon ( mc) and fluorine ( mF) atoms in atomic units
n* 1
2 3 4 5 6 7 8 9 10 a) n denotes non-exponential decay.
207
mc= 12.000 mF= 18.998
mc=oO mF= 18.998
mc= 12.00 rnpco
mC=cO mF=w
n a0 n 14 19 22 23 24 23 21 18
n n 4 15 20 22 23 23 22 21
n n 12 18 22 24 25 24 23 21
n n n 13 19 22 23 24 24 23
Case of C,( r) quasi-
208
A. Garcia-Ayllh, J. Santamaria /A two-mode model offluoroform
different from pure stretch-stretch heavy atom blocking in linear hydrocarbon chain models [ 34,35 1. The most interesting feature of trihalomethane spectra is associated with the location, splittings and intensities of the absorption bands within each polyad which allow one to quantitatively determine the bend-stretch anharmonic coupling constant, X,, and the resonance coupling constant Ksbb. Both constants include kinetic and potential contributions. At present we are studying the resonance Hamiltonian by analytical evaluation [ 36 ] of the appropriate Fourier coefficients with convenient analytical form for the kinetic and potential contribution. The potential coupling contribution was analytically intractable for the Gaussian form of attenuation [ 131, but it is not true for other cases. It will be very interesting to compare our results for resonance phase space structure with those of algebraic resonance dynamics method obtained by Kellman [ 191. The short time exponential decay rate is believed to determine the envelope of the overtone absorption line [ 9,11 ,121 for the situation of homogeneous broadening. Our results agree with the values obtained using the time evolution matrix method. Nevertheless one cannot extract quantitative conclusions, because as is well known, very large discrepancies have been found in other systems (for example in benzene) between calculated [ 12,131 and experimentally observed linewidths [ 37 1. The inclusion of all degrees of freedom for fluoroform (i.e. taking into account CF stretches and FCF bends in a curvilinear coordinate representation) and other polyatomics complicates tremendously the treatment, as Halonen [ 38 ] has recently indicated. Often, in practice, it is necessary to include only strongly coupled motions (Fermi resonances between stretch and bends) as in our case of fluoroform, and in the recent calculation of CHzClz [ 391. In other cases, it is possible to couple only levels that are in strong resonance with each other and to treat weaker couplings as perturbations, such as has been done for CHD3 [ 401. The adequacy of the two-mode model to explain the dynamics and Fermi resonance structure in the IR spectrum of fluoroform, enable us to extend this study to other molecules of the type CHX3, such as HCC&, CHD3, CH(CFs)3, etc., where the model should work better for molecules with a heavy CX3 group.
Acknowledgement This work was supported by CICYT grant No. PB86-0540. We thank Professor G.S. Ezra and Dr. B.G. Sumpter for helpful comments. It is also a pleasure to acknowledge stimulating discussions with Professor R.D. Levine. One of us (AGA) gratefully acknowledges the receipt of a FPI scholarship during the course of this work.
References [ 1] K.V. Reddy, D.F. Heller and M.J. Berry, J. Chem. Phys. 76 (1982) 2814. [2] H.R. Dtlbal and M. Quack, J. Chem. Phys. 81 (1984) 3779. [ 31 J.E. Baggott, MC. Chuang, R.N. Zare, N.R. Diibal and M. Quack, J. Chem. Phys. 82 ( 1984) 1186. [ 41 S. Peyerimhoff, M. Lewerenz and M. Quack, Chem. Phys. Letters 109 ( 1984) 563. [ 51 J.S. Wong, W.H. Green, C. Cheng and C.B. Moore, J. Chem. Phys. 86 (1987) 5994. [6] F.F. Crim, Ann. Rev. Phys. Chem. 35 ( 1984) 657; T.R. Rizzo, C.C. Hayden and F.F. Grim, Faraday Discussions Chem. Sot. 75 ( 1983) 223; J. Chem. Phys. 8 1 (1984) 4501; L.J. Butler, T.M. Ticich, M.D. Likar and F.F. Crim, J. Chem. Phys. 85 (1986) 2331; M.D. Likar, J.E. Bagot, T.M. Ticich, R.L. Vander Wal and F.F. Grim, J. Chem. Sot. Faraday Trans. II 84 (1988) 1483. [ 7 ] E.L. Sibert, J.T. Hynes and W.P. Reinhardt, J. Phys. Chem. 87 (1983) 2023. [8] J.S. Hutchinson, W.P. Reinhardt and J.T. Hynes, J. Chem. Phys. 79 (1983) 4247; J.S. Hutchinson, J.T. Hynes and W.P. Reinhardt, J. Phys. Chem. 90 (1986) 3528. [9] E.J. Heller and M.J. Davis, 84 (1980) 1999; Accounts Chem. Res. 14 (1981) 368. [lo] W.P. Reinhardt and C. Duneczky, J. Chem. Sot. Faraday Trans.1184(1988) 1511. [ 111 E.L. Sibert, W.P. Reinhardt and J.T. Hynes, J. Chem. Phys. 81 (1984) 1115,1135. [ 12 ] R.J. Wolf, D.S. Bathia and W.L. Hase, Chem. Phys. Letters 132 (1986) 493; D.H. Lu, W.L. Hase and R.J. Wolf, J. Chem. Phys. 85 ( 1986) 4422; D.H. Lu and W.L. Hase, Chem. Phys. Letters 142 (1987) 187. [ 13 ] A. Garcia-Ayllon, J. Santamaria and G.S. Ezra, J. Chem. Phys. 89 (1988) 801. [ 141 H.R. Dilbal and M. Quack, Mol. Phys. 53 (1984) 257; A. Amrein, H.R. Dtlbal and M. Quack, Mol. Phys. 56 ( 1985) 727; M. Lewerenz and M. Quack, Chem. Phys. Letters 123 (1986) 197.
A. Garcia-Aylbh, J. Santamaria /A two-mode model offruoroform [ 151 G.A. Voth, R.A. Marcus and A.H. Zewail, J. Chem. Phys. 81 (1984) 5494. { 16 ] W.H. Green, W.D. Lawrence and C. Bradley Moore, J. Chem. Phys. 86 ( 1987 ) 6000. f 171 L. Halonen, T. Carringon Jr. and M. Quack, J. Chem. Sac. Faraday Trans. II 84 ( 1988 ) 137 1, [ 18] T. Uzer, J.T. Hynes and W.P. Reinhardt, J. Chem. Phys. 85 (1986) 5791; B.G. Sumpter and D.L. Thompson, J. Chem. Phys. 82 (1985) 4557; 86 (1987) 2805. [I91 M.E. Kellman and ED. Lynch, J. Chem. Phys. 85 (1986) 7216; ME. Ke&nan, J. Chem. Pbys. 83 ( 1985) 3843; Chem. Pbys. Letters 113 (1985) 489. f 201 C.J. Cejan, S. Shi and W.H. Milfer, J. Phys. Chem. 86 (1982) 2244; S. Shi and W.H. Miller, Theoret. Chim. Acta 68 ( 1985) 1. [ 2 1 ] K.L. Bintz, D.L. Thompson and J.W. Brady, J. Chem. Phys. 85 (1986) 1848; Y. Guan, G.C. Lynch and D.L. Thompson, J. Chem. Phys. 87 (1987) 6957. [22] M. Par&an, I. Suzuki and J. Overend, J. Chem. Pbys 42 (1965) 2335; AR. Hoy, I.M. Mills and 6. Strey, Mol. Phys. 24 (1972) 1265; R. Wallace, Chem. Phys. 11 ( 1975) 189; S. Carter and NC. Handy, Computer Phys. Rept. S ( 1986) 115. [23] L. Halonen and ‘I’.Carrington, J. Chem. Phys. 88 ( 1988) 4171; T. Can-&ton, J. Chem. Phys. 86 ( 1987) 2207.
209
[24] B.V. Chirikov, Phys. Rept. 52 (1979) 265. 1251 E.B. Wilson, J.C. Decius and P.C. Cross, Molecular Vibrations (Dover, New York, 1980). 126f HR. D&al, T.K. Ha, M. Iewerenz and M. C&sack,J. Chem. Phys., in press. I27 I J.N. Murrell, in: Gas Kinetic and Energy Transfer, Voi. 3. eds. PG. Ashmore and R.J. Donovan (Cbem. Sot., London, 1978) ch, 7. 128 A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer, Berlin, 1983). I29 N. Sigrist, in: Lag-Time Predictions in Dynamics, eds, V. Szebehely and B.D. Tapley (Reidel, Dordrecht, 1976) p. 141. l3OlJ.A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems (Springer, Berlin, 1985 ). [31] R.B. Shirts, J. Chem. Phys. 85 ( 1986) 4949. [32] E.J. Weller, E.B. Stechel and M.J. Davis, J. Chem. Phys. 73 ( 1980) 4720. [ 331 A. Garcia-Ayllon and J. Santamaria, in preparation. 1341 V. Mpez and R.A. Marcus, Chem. Phys. Letters 93 (1982) 232. f35] K.N. Swamy and W.L. Hase? J. Chem. Phys. 82 (1985) 123. (36] T. Uzer, Chem. Phys. Letters 110 f I984 ) 356. f37f R,H. Page, Y.R. Shen and Y.T. Lee, Phys. Rev. Letters S9 (1987) 1293; J. Chem. Phys. 88 (1988) 4621; 88 (1988) 5362. [38] L. Halonen, J. Phys. Chem. 93 (1989) 3386. [39] L. Halonen, J. Chem. Phys. 88 (198) 7599. [40] EL. Sibert III, J. Chem. Phys. 88 (1988) 4378,
STRETCH-BEND COUPLING EFFECT ON THE INTRAMOLECULAR VIBRATIONAL RELAXATION IN A TWO-MODE MODEL OF CH OVERTONE EXCITED FLUOROFORM
Andms GARCfA-AYLLGN and Jesus SANTAMARfA Departamento de Q&mica Fisica, Fact&ad de CC. Q&micas, Universidad Complutense, 28040 Madrid, Spain Received 28 Match 1989; in final form 10 October 1989
A classical dynamic study of the effect of stretch-bend coupling and CH anharmonicity on the occurrence of Fermi resonance and intramolecular vibrational relaxation (IVR) has been carried out for a two-mode Hamiltonian proposed by Halonen et al. for CH overtone excited fluoroform. The analysis is made in terms of variations of the phase space structure and overtone relaxation rate constants, with respect to CH stretch anharmonicity and various approximations to the kinetic and potential couplings. CH bond anharmonicity is confirmed as the most important factor in the dynamics of overtone relaxation and it controls the occurrence and intensity of the 2 : 1 Fermi resonance between the CH stretch and the adjacent bends. The kinetic coupling is much more important than the potential, and, contrary to the effect of this, it tends to decrease the stability of the CH periodic orbit. The overtone relaxation rate constants, estimated by linear stability analysis of the CH periodic orbit, are in good agreement with the experimental results.
1.
Introduction
The advent of new experimental techniques, such as overtone spectroscopy has provided information about molecular behavior at large amplitude vibrational motions. Recently there has been a large amount of experimental research on the overtone spectroscopy of CH and OH local modes [ l-6 1. The dynamics of coupled vibrations is of capital importance in the understanding of molecular spectra and intramolecular energy transfer. At the same time the presence of low-order frequency commensurability between anharmonic zeroth-order modes (i.e. the generalized Fermi resonances) is the cause of the intramolecular vibrational relaxation (IVR) and splittings in the vibration-rotation spectra [ 71. Assuming that the observed linewidths are due to homogeneous broadening, the time scale of the initial vibrational relaxation can be inferred [ 8,9]. In each case the overtone relaxation is found to occur within a fraction of a picosecond [ lo]. Much recent theoretical work [ 11-211 has been focused on the extraction of information from overtone spectroscopy on the specific pathways that govem IVR. Of particular importance is the coupling between CH (or OH ) stretching vibrations and adjacent 0301-0104/90/$03.50 ( North-Holland )
0 Elsevier Science Publishers B.V.
bending modes. The initial energy transfer from the excited stretch to the adjacent modes has been shown to proceed via a 2 : 1 Fermi resonance whose magnitude varies with the specific nature of each molecule. Several resolved absorption bands appear in each overtone region in trihalomethanes [ 2-51. These bands result from a very strong coupling between the CH stretch and the bends and are characteristic of Fermi resonances. The measured splitting within each polyad allows the stretch-bend anharmonic coupling to be determined quantitatively, and demonstrates that CH stretching energy flows first to the bending modes and, subsequently, to the heavy atom vibrations on a longer time scale. It has been shown [ 2-5 ] that these Fermi resonances in CHFs, CHD, and CH( CF,)3 can be analysed via effective tridiagonal Hamiltonian matrices. The diagonal terms contain both harmonic and anharmonic contributions, while the offdiagonal Fermi resonance coupling matrix elements depend on an effective coupling constant (KW). The use of curvilinear internal coordinates and their appropriate symmetrized combinations [ 7,16,22,23 ] represents clear advantages over the Cartesian normal mode coordinates for the large amplitude vibrational motions of the overtone excitations. First of
198
A. Garcia-Aylldn, J. Santamaria / A two-mode model ofjuoroform
all, it accounts for the anharmonicity of X-H bonds; secondly, curvilinear coordinates give a more accurate representation of the potential energy surface when compared with rectilinear coordinates, and, tinally, the potential surface parameters are isotope independent within the Born-Oppenheimer approximation. A problem with these coordinates is the fact that the G matrix elements of the kinetic energy are much more complicated than in the case of rectilinear coordinates. The study of Fermi resonances between CH stretching and HCF bending vibrations of fluoroform has been carried out by Halonen et al. [ 171, using a Hamiltonian described in terms of symmetrized curvilinear internal coordinates and their conjugate momenta. The kinetic energy operator is obtained by expanding the G matrix in terms of the Morse function y = 1 - exp ( - PAr ) , and retaining only the relevant terms for the Fermi resonance problem. The potential energy function is also expanded about the equilibrium configuration in terms of y. Symmetrized internal combinations of curvilinear coordinates were used to reduce the problem to a two-mode (HC stretch and a doubly degenerate bend) model Hamiltonian. These coordinates were primarily used by Voth et al. [ 151 in their theoretical analysis of the highly excited CH vibration in partially deuterated methanes. In this study, the G matrix elements of the Hamiltonian were evaluated with all internal coordinates at their equilibrium values, except the CH coordinate r. In addition to these G matrix couplings, higher-order (i.e. higher than quadratic) curvilinear potential energy terms were needed in order to explain more quantitatively the observed spectra of deuterated methanes. Halonen et al. [ 171 use the same type of approach but they expand the kinetic energy operator in terms of y. The model Hamiltonian they use gives band positions and relative band intensities within polyad spectra with enough accuracy. At the same time, the description of Fermi resonance is made in terms of force constants which have a precise definition. A detailed dynamical treatment, but restricted to the effective resonance Hamiltonian spectroscopically fitted, has been carried out by Kellman [ 19 1. The underlying idea of this approach, so-called algebraic resonance dynamics, is that every element of the spectroscopic fit corresponds to an element of the non-linear dynamics. The polyad phase space pro-
files obtained by Kellman for CHX3 molecules render a phase space structure very different from the simple pendulum resonance Hamiltonian model 1241. In this paper, we present a classical dynamical treatment of the effect of both kinetic and potential couplings between the CH stretch and the doubly degenerate bend for several variations of the two-mode model Hamiltonian proposed by Halonen et al. [ 17 ] for CHF3 molecule. Following the treatment used in our study of a two-mode model of benzene [ 13 1, we are trying to understand the sensitivity of short time IVR to potential and kinetic couplings in terms of the classical phase space structure of the model Hamiltonian for fluoroform. The change from stability to instability (and vice versa) of the anharmonic CH periodic orbit and the variation in intensity of the Fermi resonance between the stretch and the bends are a function of potential and kinetic coupling as well as stretch and bend anharmonicities. It is our goal to study the changes in phase space structure (and Fermi resonance intensity) and the variation of overtone relaxation rate constants with respect to various approximations of both kinetic and potential coupling terms of the two-mode Hamiltonian. The various versions of the Hamiltonian are described in section 2, where we discuss the approximations we use. In section 3, the Poincart surface of section is used to examine changes in phase space structure induced by the stretch-bend potential and kinetic energy couplings. Section 4 is devoted to estimate the short time overtone relaxation by the method of linear stability analysis. In section 5 we present the conclusions and discuss possibilities for future work.
2. Model Hamiltonian The reduced classical Hamiltonian used to study the Fermi resonances between the CH stretch and the adjacent HCF bending vibrations in fluoroform CHCFJ is described in terms of internal coordinates and conjugate momenta as follows (see fig. 1):
A. Garcia-Ayllbn, J. Santamaria / A two-mode model offluoroform
199
where t,uis the dihedral angle defined by (Yand #. Taking advantage of the symmetry of the problem, the Hamiltonian (2.la) is transformed to symmetrized internal coordinates [ 17 ] ‘%= 3-l”
(pi
+v)2
6 = 6-“‘2 (2% 62=
Fig. 1.Molecular model of CHF3.
2-l’*
+cQjh
-92
-$@3),
(2.3a) (2.3b) (2.3~)
(v)z-g3),
and their conjugate momentap,, p6,, plh. If the totally symmetric bending coordinate 0, is fixed at its equilibrium value, then its conjugate momentum ps disappears, and the Hamiltonian takes the form
+ Ge,& pet P& + i=q 2 Grei PC~ei +D[
1-exp( -/3Ar)12+ V(r, 8,, tl,),
(2.4)
where V(r, 8,, 6,) can be taken as a Taylor expansion similar to (2.1 b) but now in terms of or, S, and r. The new G matrix coefficients are
(2.lb) Here (r, p,) are the coordinate and momentum of CH stretch, and (pi, pe i= 1, 2, 3) are the three HCX bending vibrations. The CH stretch is described by a Morse oscillator and the bends can be considered as harmonic oscillators with r-dependent force constants as we will show later on. The CF bond stretch coordinates and the FCF bond angle a! are frozen at their equilibrium values (R=1,332~,anda=108” respectively,seetig. 1). The kinetic energy G-matrix elements are [ 25 ] (2.2a)
G,=lImH+lImc,
- 2 cos &rR)/mc,
G,j = -sin ~i~Rm~,
[ cos
W/r2- 2 cos
(-G22+G33+G,2-G,3),
G&,= 6-1’2 (2G,, -GR-Gr3), G&z
2-lf2 (Gr2-G,3).
(2.5)
The approximation made following Voth et al. [ 15 ] is to treat the dynamical G matrix elements in eqs. (2.2) as functions of the CH stretch coordinate exclusively, with the rest of the variables (R, pi, v/) at their equilib~um values (see table 1) . Consequently the G matrix elements in terms of pior B are G22
=G33
=G,(r),
(2.2b)
GEz=G t3=G23=Gwstr),
(2.2c)
G,,,,=G,,=G,(r)=G,(r)-Go,, GBI&-0 9
Gti=cos ylr2mH +
G@,&= 3-l”
GI I =
Gii = 1/r2mH+ 1/R2mc +(l/r2+1/R2
G.a=t~22+fG33-G23,
c;Oicos
G,.o,=Gti=G,+=O.
y/rR
+ (sin’pli sin2yl+cos (Ycos y)/R2]/mc,
(2.2d)
(2.6)
Since this is the most obvious simplification of the
200
A. Garcia-Aylh. J. Santamanh /A two-mode model ofJluoroform
Table 1 Surface parameters
H= ;G,p;+; Ab initio
f K) &A) a=~+-
(deg)
P=PW
(deg)
Fit
39386 1.861 1.098 I .332 108.0 110.13 1.008 18.998 12.000 44540 - 18997 - 88895
D (cm-‘)
mH (au) mF (au) mc (au)
(cm-’ radm2) F, (cm-’ A-’ radm2) F,, (cm-’ A-* rade2) F
jG,
42310 1.801
+D[
40854 10550 - 284000
H=tG,,p:+fGBepi+D[l-exp(-j?Ar)]’
J
P;
V(r, 8,, 0,).
The effect of the cross term in p,pe is very small due to the small numerical values of G2 given by Halonen et al. [ 171. With respect to the angle dependence of the potential function V( r, 0,, 0,) it is convenient to take the bending vibrations as harmonic oscillators with r-dependent force constantfO( r) in the form:
= t LLi(rM+Mrb%l
= i.L4r)e2,
fe(r)=Fw+
Halonen et al. [ 17 ] have carried out an expansion of the “quasiexact” Geeand Gti matrix elements in terms of the Morse function y= 1-exp( - /IAr) and of 0, respectively, resulting in
Gti,==G$i+G2Oi
f (i=l,
>
$G3+fGl
~J’,wY
2)y
(2.8)
Ar=O, p=O), GS = (aZG&r2)er
(2.11)
Alternatively this is easily verified by expanding the potential function in (pand 8 variables, Fee =fap--fwf>
Frm=fw-fw,, Fwee=fr,,-f-p.
Y2,
where
G, = (6IGeel%)e,
(2.10)
with e2= f3: + 0:. The r dependence of fe is easily found by expanding in terms of y ( Morse function )
y2.
jG,y+
(2.9)
12.7)
e2),
P; =PW +&.
G&=G,(
>
Y’
1 -exp( -/3Ar)]‘+
with
G,=G&+
1
1
problem, we will consider the expressions for GBBand G,+given in (2.6) as quasiexact. The classical Hamiltonian now takes the form
+ Ur, 4,
G&+ ;GY
(2.12)
Here the f symbols stand for the coefficients up to fourth order in the expansion of the potential in r and $ displacements. In this way we have an analytical expression for the stretch-bend potential coupling in terms of a switching function S(y) of the form Ur, 0) = t%Y)&@,
G~,=G~=G~(Ar=O,yl=O)=O, G2 = (aGJ@),. In this approximation the classical Hamiltonian takes the form
(2.13) The switching function in (2.13) is a parabola in y which depends on the values of the potential expan-
201
A. Garcia-Ayll&n,J. Santamaria / A twcFmodemodel offluoroform
sion coefficients. Two series of coefficient values are given by Halonen et al. (see table 1 ), the first coming from ab initio calculations [ 26 1, and the second obtained by fitting the values of FM and F,* (and consequently the parameters X, and Ksbbof the diagonal and tridiagonal elements [ 21) to the experimental spectrum up to the chromophore quantum number of the polyad N= n,+ in@= 6 (for A, states) and 1 1/ 2 (for E states). The analytical expressions for S(y) for both series of parameters are S(y) = l-O.229 ly- 0.4027y2,
(2.14a)
S(y) = 1+0.1434y-o.9999y2,
(2.14b)
respectively, and the representations are given in fig. 2 within the range from y= - 1 (Ar= -0.37 A) to y= 1 (At= + 00). Both functions differ considerably for Ar < 0, and at the same time, neither give the correct asymptotic behavior at y= 1, S( 1) =O. In order to correct this anomaly we have used an analytical expression of the form [ 27 ] S(y) = 1-tanh(P&) (2.15)
=2(1-y)*‘/[l+(l-y)2c],
which gives the asymptotic behavior and passes for (co.3 16 through the two crossing points of the switching functions previously indicated (see fig. 2). I
-
Ab m,t,o
-,-
Asymptotac
In fig. 2 the classical turning points of the CH Morse function for n,= 4 and 8 are also shown. In summary, we have different expressions for the two-mode (CH stretch and a doubly degenerate bend) model Hamiltonian, which include two forms ( (2.7) and (2.9) ) for the kinetic energy and four for the potential coupling: two coming from (2.13) with “ab initio” and “fitted” potential parameters respectively ( 2.14 ) , the asymptotic expression ( 2.15 ) and finally the case of no potential stretch-bend coupling, i.e. S(y) = 1. All possible combinations allow us to study in depth the effects of changing contributions of the kinetic and potential coupling on the stability of HC periodic orbit and on the intensity of 2 : 1 Fermi resonance.
3. Stretch-bend potential, kinetic couplings and phase space structure We have used surfaces of section [ 281 to study the phase space structure of the two-mode model Hamiltonian for the various versions previously described. Each surface of section (SOS) is obtained by plotting (f3,pe) every time r= r. and pr> 0 for several trajectories at a particular constant energy and different initial actions. The gross features of the phase space corresponding to a system of two degrees of freedom with the possibility of 2 : 1 Fermi resonance are already well known [29,30,13]. In our system, however, the anharmonic stretch-bend resonance is known to be small for the stretching fundamental and the first two overtones, but the large Morse anharmonicity of the CH stretch can enhance the resonance. In fact, the Morse frequency o, is defined as o, =a( l -&!1,/2D),
(3.la)
where Q is the harmonic frequency, 9= (2DB* x Grr)“‘, and Z, is the Morse action, while the bend harmonic frequency oh with an r-dependent force constant S( y ) Fm has the expression we= ]GBBS(Y)J’~~I”~,
r.-0.37
A
Y Fig. 2. Switching fimctions S(y).
(3.lb)
with the result that w, can be larger, smaller or approximately equal to two times o& Representative SOSs for quasiexact Gee and asymptotic potential coupling ( 2.15 ) are shown in fig. 3 at energies corresponding to initial states (n, ne=O), n,=2, 3, 5 and
202
A. Garcia-Ayllbn. J. Santamaria /A two-mode model ofJluoroform
3.00
liil,i,,,,,,,,,,,,,,,,,l,,,,,,,,,,,,l,,,,,,,
d i
Fig. 3. SOS for Hamiltonian (2.7) ( and asymptotic S(y) at energies corresponding to overtone excited states (n, ne). Representation ofp, (in amu A/IO-” s) versus 19(in rad). (a) State (2,0), (b) state (3,0), (c) state (5,0), (d) state (10, 0).
10. At n,= 2, the structure of SOS is very simple with a stable (elliptic [ 281) central fixed point, because we are far from resonance. In fig. 3a the SOS is not shown in full: a more detailed representation should give us two small islands at the top left and bottom right [ 291, such as are observed for nr= 5 (fig. 3~). There is a single family of regular trajectories (apart from microscopic regions of localized stochasticity which are always present in non-integrable systems [ 281) continuously spanning the range between two
limiting periodic orbits: the pure CH stretch motion, corresponding to the central fixed point in the SOS, and the periodic orbit piercing the SOS at the center of each of the two islands [ 131. Fig. 3b shows the SOS of the lower energy level in which the central fixed point is unstable (hyperbolic with reflection [28] ) with two elliptic points clearly visible. Obviously we are getting close to the resonance. Again the two islands mentioned in the previous cases are not shown. The situation is more clear in fig. 3c where
A. Garcia-Ayl~~m~ f. Santamark /A two-mode modei of~uor~forrn
there are two families of trajectories divided by the separatrix of the resonance: one resonant family (small region inside the separatrix) with large interchange of energy between the stretch and the bend, and other non-reson~t family (large region outside the separatrix). A representative trajectory of the resonant family is drawn in fig. 4a. The two islands at the top left and
1
.oo i
/
a
$
500
-f.OO C
j
E
0.00 ?
3 -1.50
m--lo
0.00
1.25 CH
Fig. 4. Can~guration space plots. Rep~~n~tion off? (in rad) versus r (in A). (a} Representative trajectory of the resonance family at the (4,O) state for Hamiltonian (2.7) and asymptotic S(y). (b) Same for state (8,O).
203
bottom right of the SOS of fig. 3c correspond to a single trajectory of the non-resonant family, and are a consequence of the particular cut used to define the SOS [ 13 1. At n,= 7 and 8 we are exactly in resonance (with an hyper~lic central fixed point and four elliptic ones) and, as a result of the bifurcation of the CH stretch periodic orbit, we have two families of resonant trajectories divided by a unique separatrix (see fig. 5d). The first family is delimited by the two elliptic points at the top right and bottom left and by the separatrix, and representative trajectories have the form of ( (open right) in the configuration space plot (r, 8), similar to fig. 4a. They correspond to the situation where stretch elongation and angle increment are in phase. The second, also resonant family, delimited by the two other elliptic points and the separatrix, define trajectories where stretch elongation and angle variation are in phase opposition. Representative trajectories have the form of > (open left) (see fig. 4b). As it is clear from fig. 5d, the ( family gives more interchange of energy between stretch and bend due to the different variation ofp, in the two limiting stable trajectories piercing the SOS at their couple of elliptic points. When the energy of CH increases (higher overtones) the system goes out of resonance due to the CH strong anharmonicity. For example, at nr= 10 (fig. 3d) we have a situation similar to the case n,=4 with two families of trajectories (resonant and nonresonant), but now the resonant family includes the trajectories of the form of) (i.e. with r and 8 motions in phase opposition). On the other hand, the correlation between the cases n,=5 and t2,= 10 cannot be complete, because in the last case we are approaching stochasticity. As we mentioned in section 2, four types of potential coupling have been investigated. The effect of the potential coupling is not very severe within the range of bend attenuation studied. There are, nevertheless, significant differences which deserve some attention. As is well known [ 12,13 ] the potential coupling reduces the range (in energy) of instability in the CH stretch periodic orbit. Within the range of energies explored (0 < n,< lo), the strongest potential couplings correspond to the cases of asymptotic and ab initio switching functions, because they give the smallest resonance width with the closest distance in phase space between the two elliptic resonant points (see fig. 5a for n,=4). The asymptotic switching
-
2
31
1
I
/
,
I
,
,
,
a
0.00
-3.00 -;
2
0.00 -
-3.00
pi -2.00
0.00
2 00
3.00
Fig.S.SOSforHamiltonian (2.7).Representationofp~(inamu~2/10-’4s)versusB(inmd). (a) State (4,0)andabinitioS(y), (b) state (4,O) and fitted s(y), (c) state (4,O) and no coupling, (d) state (8,O) and ab initio S(y), (e) state (8,O) and fitted S(y), (f) state ( 8,0 f and no coupling.
A. Garcia-Ayllbn, J. Santamaria / A two-mode model offluoroform
function yields a slightly lower instability than the ab initio function. The empirically fitted attenuation function (fig. 5b) offers an anomalous behavior, in the sense that, although the resonance width is lower than in the case of no coupling (and larger than in the two previous cases), see fig. 5c, nevertheless, the pure resonant trajectory gives more interchange of energy (i.e. larger distance between elliptic points than in the no coupling situation). The reason is due to the fact that S(y) can be greater than unity for values of y slightly positive, while for y
205
0
2 0.00-
-3.00 !I -2.00
. ‘., I’,:i : .‘.J,y-“‘i .\ .I~ ;.,>..,‘., , .,, ‘f ., ::.: +_.,__ :,+ *,.,, .’1;:;.:...‘,:. : .‘i . “.“’ ’ : ,i : . . .‘% . . I ._... -I :: ‘..T..::: 4 ” ...s .,::...i -.<..’ . : _... ,:,.-... ,i, ,. :.:A:,.., .., . .._ I.. . ;f,,‘,” - “.$$ .i’ :, ‘:,‘. “..:,... >:.,.. : . +.;..,,,.i:;. /X!‘, -...I,_
000
2 00
19
Fig. 6. SOS for Hamiltonian (2.9) with ab initio potential coupling at different overtone excited state (n, ne) . Representation ofp, (in amu A2/10-” s) versus 0 (in rad). (a) State (4, 0), (b) state (8,O).
4. Short time overtone relaxation and linear stability analysis The short time decay rate constants from the CH overtones in the two-mode model can be estimated very accurately by linearization of the dynamics [ 28 ]
A. Garcia-Ayiih, J. Santamarfa /A two-mode model offluoroform
206
around the HC periodic orbit [ 13,321. Linear transformations are area preserving and there is a Jacobian (2 x 2 ) matrix propagator whose determinant is unity. The eigenvalues of the Jacobian matrix propagator of the return map are I, =exp( ia), R,=exp( -irr) for elliptic (stable) fixed point, and A,=exp(cr), Az=exp(-o) for hyperbolic (unstable) fixed point (D real), where the exponents are the (real or complex) phases acquired by the eigenvectors after 4 (the period of the orbit). In the unstable case it is possible to estimate the decay constant k for the stretch-bend relaxation, since then rr=l~7~,. The method requires the numerical calculation of the Jacobian matrix by determining the action of the return map on initial conditions displaced by small amounts (Ape, AfI=O.OOl ) along the 0 and pe directions respectively. This is easily achieved by running for one period r, a couple of trajectories, with initial conditions displaced the first trajectory in pB _ and the second in 0. The relaxation rate k is estimated by diagonalizing the resulting Jacobian matrix. The values of k (ps-’ ) for all the situations discussed in section 3 are given in table 2. The comparison of the values of k confirms all the deductions we made in terms of phase space structure. Only the homogeneous widths (r,,,) can be associated with the decay of the population for initially prepared CH stretching and bending states (n,, n,; nr= 1, .-‘> 10, ne=O). For CHF, the exponential decay time 7= 1/2~cr,,~, was estimated, using the
time evolution matrix method [ 21, to be about 0.1 ps for the state n,= 3. The value we get for the same initial state is about 0.07 ps (r=75 cm-’ ), and for n,=8 t is of the order of 0.04 (r= 130 cm-‘). For CH f CF3) 3 Baggott et al. [ 3 ] gave rough estimates for lY which are between f= 7 cm-r for the N= 2 polyad and r=70 cm-’ for the N=6 polyad, these widths are related to decay times which range between 0.8 and 0.08 ps. Our results are reasonable and fall within the experimental uncertainties. It is interesting to note the slight effect of the potentiai coupling on the stability and resonance of the periodic CH orbits. In fact, the suppression of the coupling displaces the resonance toward lower overtones (from n,= 7, 8 toward n,= 5, 6 ) and increases the decay constant by less than a factor of two. Much more pronounced changes appear by suppressing the kinetic coupling, i.e. taking r equal to (r) of each overtone level in the Gee expression (G( (r) ) zconstant). In that situation the instability and resonance are only present within the range 2
Table 2 Rate constants (ps-’ ) for overtone decay in model Hamiltonian for fluoroform from linear stability analysis %
I 2 3 4 5 6 7 8 9 10
Fitted S(y)
Asymptotic S(y)
Ab initio S(y)
No potential coupling
G, approx.
GMquasiexact
GBB constant
GIXI approx.
GBequasiexact
GBB approx.
Gwquasiexact
Gm approx.
GBBquasiexact
n a) 4 16 21 23 24 23 21 18 12
n n 14 19 22 23 24 23 21 18
n 7 12 11 n n n n n n
1: 25 30 32 33 34 33 32 30
n 16 24 28 31 33 33 33 33 32
n 5 17 22 25 27 27 26 24 20
n n 15 21 24 26 27 27 26 24
16 27 34 37 38 38 35 30 21 n
14 26 33 36 38 38 37 34 29 21
a1 n denotes non-exponential decay.
A. ~ar~ia-Ay~l~~,.I. Santamaria / A two-mode model of~~oroform
effects work in the opposite direction: the potential coupling produces stability of the CH periodic orbit, while the kinetic renders the system more dynamically unstable. A quantitative study of those effects is under way [ 331, where we treat the stretch-bend resonance Hamiltonian by Fourier expansion, via the analytical evaluation of the appropriate Fourier coefficients [ 13 1. We have also investigated the question of heavy atom blocking on stretch-bend vibrational energy transfer for this model of fluoroform. Working with infinite mass for carbon atom, for fluorine atom or for both atoms, the modifications on the stability decay of CH periodic orbit in comparison with the case of real atom masses are very small as one can see in table 3. This is not surprising since the specific nature of a bend motion prevents an easy blocking of energy transfer from the adjacent stretch.
5. Discussion and remarks In this work we have examined the sensitivity of classical dynamics of a two-mode model Hamiltonian for fluoroform. By examining the classical phase structure of the two-mode system, we have shown that there is a strong correlation between exponential short time decay constant by using linear analysis and stability of the CH stretch periodic orbit as observed in the phase space structure. In a previous work [ 131 we showed that the values of the relaxation rate con-
stant obtained by linear stability analysis were in good agreement with the quasiclassical trajectory results as measured from the probability P*(t) for remaining in the initial state (n, 0). This last finding, which we assume to be general, dispenses with running batches of quasiclassical trajectories in this study. The effect of CH bond anha~onicity on the stability and resonance of the CH periodic orbit has been shown to be, as in similar systems [ 1l-l 3 1, extremely important. On the other hand, the effect on the dynamics as coming from changes in the potential coupling is not very strong within the explored range of stretch-bend attenuation. One would need a substantial change (much more strong attenuation [ 12 ] ) to produce significant changes on the dynamics, such as changes from instability to stability or from resonance to out of resonance of the periodic orbits. Nevertheless the effects are noticeable and one can correlate the increase of potential coupling attenuation with the increase of stability of the CH periodic orbit. The suppression of the kinetic coupling produces a dramatic effect on the dynamics and phase space structure of the two-mode system. Again one can correlate, contrary to the potential coupling effect, the increase in strength of the kinetic coupling with the decrease of stability of the orbits (and with the displacement of the resonance to lower overtones ), We have also confirmed that the heavy atom blocking effect does not exist for stretch-bend vibrational energy transfer. This situation is nevertheless
Table 3 Heavy atom blocking effect on the rate constants fps-’ ) for overtone decay in model Hamiltonian of ~uo~fo~. exact and asymptotic S’(y). Masses of carbon ( mc) and fluorine ( mF) atoms in atomic units
n* 1
2 3 4 5 6 7 8 9 10 a) n denotes non-exponential decay.
207
mc= 12.000 mF= 18.998
mc=oO mF= 18.998
mc= 12.00 rnpco
mC=cO mF=w
n a0 n 14 19 22 23 24 23 21 18
n n 4 15 20 22 23 23 22 21
n n 12 18 22 24 25 24 23 21
n n n 13 19 22 23 24 24 23
Case of C,( r) quasi-
208
A. Garcia-Ayllh, J. Santamaria /A two-mode model offluoroform
different from pure stretch-stretch heavy atom blocking in linear hydrocarbon chain models [ 34,35 1. The most interesting feature of trihalomethane spectra is associated with the location, splittings and intensities of the absorption bands within each polyad which allow one to quantitatively determine the bend-stretch anharmonic coupling constant, X,, and the resonance coupling constant Ksbb. Both constants include kinetic and potential contributions. At present we are studying the resonance Hamiltonian by analytical evaluation [ 36 ] of the appropriate Fourier coefficients with convenient analytical form for the kinetic and potential contribution. The potential coupling contribution was analytically intractable for the Gaussian form of attenuation [ 131, but it is not true for other cases. It will be very interesting to compare our results for resonance phase space structure with those of algebraic resonance dynamics method obtained by Kellman [ 191. The short time exponential decay rate is believed to determine the envelope of the overtone absorption line [ 9,11 ,121 for the situation of homogeneous broadening. Our results agree with the values obtained using the time evolution matrix method. Nevertheless one cannot extract quantitative conclusions, because as is well known, very large discrepancies have been found in other systems (for example in benzene) between calculated [ 12,131 and experimentally observed linewidths [ 37 1. The inclusion of all degrees of freedom for fluoroform (i.e. taking into account CF stretches and FCF bends in a curvilinear coordinate representation) and other polyatomics complicates tremendously the treatment, as Halonen [ 38 ] has recently indicated. Often, in practice, it is necessary to include only strongly coupled motions (Fermi resonances between stretch and bends) as in our case of fluoroform, and in the recent calculation of CHzClz [ 391. In other cases, it is possible to couple only levels that are in strong resonance with each other and to treat weaker couplings as perturbations, such as has been done for CHD3 [ 401. The adequacy of the two-mode model to explain the dynamics and Fermi resonance structure in the IR spectrum of fluoroform, enable us to extend this study to other molecules of the type CHX3, such as HCC&, CHD3, CH(CFs)3, etc., where the model should work better for molecules with a heavy CX3 group.
Acknowledgement This work was supported by CICYT grant No. PB86-0540. We thank Professor G.S. Ezra and Dr. B.G. Sumpter for helpful comments. It is also a pleasure to acknowledge stimulating discussions with Professor R.D. Levine. One of us (AGA) gratefully acknowledges the receipt of a FPI scholarship during the course of this work.
References [ 1] K.V. Reddy, D.F. Heller and M.J. Berry, J. Chem. Phys. 76 (1982) 2814. [2] H.R. Dtlbal and M. Quack, J. Chem. Phys. 81 (1984) 3779. [ 31 J.E. Baggott, MC. Chuang, R.N. Zare, N.R. Diibal and M. Quack, J. Chem. Phys. 82 ( 1984) 1186. [ 41 S. Peyerimhoff, M. Lewerenz and M. Quack, Chem. Phys. Letters 109 ( 1984) 563. [ 51 J.S. Wong, W.H. Green, C. Cheng and C.B. Moore, J. Chem. Phys. 86 (1987) 5994. [6] F.F. Crim, Ann. Rev. Phys. Chem. 35 ( 1984) 657; T.R. Rizzo, C.C. Hayden and F.F. Grim, Faraday Discussions Chem. Sot. 75 ( 1983) 223; J. Chem. Phys. 8 1 (1984) 4501; L.J. Butler, T.M. Ticich, M.D. Likar and F.F. Crim, J. Chem. Phys. 85 (1986) 2331; M.D. Likar, J.E. Bagot, T.M. Ticich, R.L. Vander Wal and F.F. Grim, J. Chem. Sot. Faraday Trans. II 84 (1988) 1483. [ 7 ] E.L. Sibert, J.T. Hynes and W.P. Reinhardt, J. Phys. Chem. 87 (1983) 2023. [8] J.S. Hutchinson, W.P. Reinhardt and J.T. Hynes, J. Chem. Phys. 79 (1983) 4247; J.S. Hutchinson, J.T. Hynes and W.P. Reinhardt, J. Phys. Chem. 90 (1986) 3528. [9] E.J. Heller and M.J. Davis, 84 (1980) 1999; Accounts Chem. Res. 14 (1981) 368. [lo] W.P. Reinhardt and C. Duneczky, J. Chem. Sot. Faraday Trans.1184(1988) 1511. [ 111 E.L. Sibert, W.P. Reinhardt and J.T. Hynes, J. Chem. Phys. 81 (1984) 1115,1135. [ 12 ] R.J. Wolf, D.S. Bathia and W.L. Hase, Chem. Phys. Letters 132 (1986) 493; D.H. Lu, W.L. Hase and R.J. Wolf, J. Chem. Phys. 85 ( 1986) 4422; D.H. Lu and W.L. Hase, Chem. Phys. Letters 142 (1987) 187. [ 13 ] A. Garcia-Ayllon, J. Santamaria and G.S. Ezra, J. Chem. Phys. 89 (1988) 801. [ 141 H.R. Dilbal and M. Quack, Mol. Phys. 53 (1984) 257; A. Amrein, H.R. Dtlbal and M. Quack, Mol. Phys. 56 ( 1985) 727; M. Lewerenz and M. Quack, Chem. Phys. Letters 123 (1986) 197.
A. Garcia-Aylbh, J. Santamaria /A two-mode model offruoroform [ 151 G.A. Voth, R.A. Marcus and A.H. Zewail, J. Chem. Phys. 81 (1984) 5494. { 16 ] W.H. Green, W.D. Lawrence and C. Bradley Moore, J. Chem. Phys. 86 ( 1987 ) 6000. f 171 L. Halonen, T. Carringon Jr. and M. Quack, J. Chem. Sac. Faraday Trans. II 84 ( 1988 ) 137 1, [ 18] T. Uzer, J.T. Hynes and W.P. Reinhardt, J. Chem. Phys. 85 (1986) 5791; B.G. Sumpter and D.L. Thompson, J. Chem. Phys. 82 (1985) 4557; 86 (1987) 2805. [I91 M.E. Kellman and ED. Lynch, J. Chem. Phys. 85 (1986) 7216; ME. Ke&nan, J. Chem. Pbys. 83 ( 1985) 3843; Chem. Pbys. Letters 113 (1985) 489. f 201 C.J. Cejan, S. Shi and W.H. Milfer, J. Phys. Chem. 86 (1982) 2244; S. Shi and W.H. Miller, Theoret. Chim. Acta 68 ( 1985) 1. [ 2 1 ] K.L. Bintz, D.L. Thompson and J.W. Brady, J. Chem. Phys. 85 (1986) 1848; Y. Guan, G.C. Lynch and D.L. Thompson, J. Chem. Phys. 87 (1987) 6957. [22] M. Par&an, I. Suzuki and J. Overend, J. Chem. Pbys 42 (1965) 2335; AR. Hoy, I.M. Mills and 6. Strey, Mol. Phys. 24 (1972) 1265; R. Wallace, Chem. Phys. 11 ( 1975) 189; S. Carter and NC. Handy, Computer Phys. Rept. S ( 1986) 115. [23] L. Halonen and ‘I’.Carrington, J. Chem. Phys. 88 ( 1988) 4171; T. Can-&ton, J. Chem. Phys. 86 ( 1987) 2207.
209
[24] B.V. Chirikov, Phys. Rept. 52 (1979) 265. 1251 E.B. Wilson, J.C. Decius and P.C. Cross, Molecular Vibrations (Dover, New York, 1980). 126f HR. D&al, T.K. Ha, M. Iewerenz and M. C&sack,J. Chem. Phys., in press. I27 I J.N. Murrell, in: Gas Kinetic and Energy Transfer, Voi. 3. eds. PG. Ashmore and R.J. Donovan (Cbem. Sot., London, 1978) ch, 7. 128 A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer, Berlin, 1983). I29 N. Sigrist, in: Lag-Time Predictions in Dynamics, eds, V. Szebehely and B.D. Tapley (Reidel, Dordrecht, 1976) p. 141. l3OlJ.A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems (Springer, Berlin, 1985 ). [31] R.B. Shirts, J. Chem. Phys. 85 ( 1986) 4949. [32] E.J. Weller, E.B. Stechel and M.J. Davis, J. Chem. Phys. 73 ( 1980) 4720. [ 331 A. Garcia-Ayllon and J. Santamaria, in preparation. 1341 V. Mpez and R.A. Marcus, Chem. Phys. Letters 93 (1982) 232. f35] K.N. Swamy and W.L. Hase? J. Chem. Phys. 82 (1985) 123. (36] T. Uzer, Chem. Phys. Letters 110 f I984 ) 356. f37f R,H. Page, Y.R. Shen and Y.T. Lee, Phys. Rev. Letters S9 (1987) 1293; J. Chem. Phys. 88 (1988) 4621; 88 (1988) 5362. [38] L. Halonen, J. Phys. Chem. 93 (1989) 3386. [39] L. Halonen, J. Chem. Phys. 88 (198) 7599. [40] EL. Sibert III, J. Chem. Phys. 88 (1988) 4378,